Báo cáo toán học: "Colored trees and noncommutative symmetric functions" pdf

Colored trees and noncommutativesymmetric functions Matt Szczesny Department of Mathematics Boston University, Boston MA, USA szczesny@math.bu.edu Submitted: Oct 16, 2009; Accepted: Mar

Colored trees and noncommutative

symmetric functions

Matt Szczesny

Department of Mathematics Boston University, Boston MA, USA szczesny@math.bu.edu Submitted: Oct 16, 2009; Accepted: Mar 28, 2010; Published: Apr 5, 2010

Abstract Let CRFS denote the category of S-colored rooted forests, and HCRFS denote its Ringel-Hall algebra as introduced in [6] We construct a homomorphism from a

K+

0 (CRFS)–graded version of the Hopf algebra of noncommutative symmetric func-tions to HCRF S Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a K0+(CRFS)–graded version of the algebra of quasisymmetric functions This homomorphism is a refinement of one considered by W Zhao in [9]

1 Introduction

In [6] categories LRF, LF G of labeled rooted forests and labeled Feynman graphs where constructed, and were shown to possess many features in common with those of finitary abelian categories In particular, one can define their Ringel-Hall algebras HLRF, HLFG

If C is one of these categories, HC is the algebra of functions on isomorphism classes of C, equipped with the convolution product

f ⋆ g(M) := X

A⊂M

f (A)g(M/A), (1.1)

and the coproduct

∆(f )(M, N) := f (M ⊕ N), (1.2) where M ⊕ N denotes disjoint union of forests/graphs Together, the structures 1.1 and 1.2 assemble to form a co-commutative Hopf algebra, which was in [6] shown to be dual

to the corresponding Connes-Kreimer Hopf algebra ([5], [2]) In [6], we also defined the Grothendieck groups K0(C) for C = LRF, LF G and showed that HC is naturally graded

by K0+(C) - the effective cone inside K0(C)

From the point of view of Ringel-Hall algebras of finitary abelian categories, the charac-teristic functions of classes in K0+ are interesting If A is such a category, and α ∈ K0+(A),

we may consider κα - the characteristic function of the locus of objects of class α inside Iso(A) (for a precise definition, see [4]) It is shown there that the κα satisfy

∆(κα) = X

α 1 +α 2 =α

α 1 ,α 2 ∈K0+(A)

κα 1 ⊗ κα 2 (1.3)

In this note, we show that these identities hold also when A is replaced by the category CRF of colored rooted forests If S is a set, and CRFS denotes the category of rooted forests colored by S, we show that K0(CRFS) = Z|S|, and if α ∈ K0+(CRFS), we may define

κα := X

A∈Iso(CRF S ) [A]=α

δA

i.e the sum of delta functions supported on isomorphism classes with K-class α We show that the κα satisfy the identity 1.3

As an application, we construct a homomorphism to HCRF S from a K0+(CRFS)–graded version of the Hopf algebra of non-commutative symmetric functions (see [3]) More pre-cisely, let NCCRF S denote the free associative algebra on generators Xα, α ∈ K0+(CRFS),

to which we assign degree α We may equip it with a coproduct determined by the requirement

∆(Xα) := X

α 1 +α 2 =α

α 1 ,α 2 ∈K +

0 (CRF S )

Xα 1 ⊗ Xα 2,

with which it becomes a connected graded bialgebra, and hence a Hopf algebra We may now define a homomorphism

ρ : NCCRF S → HCRF S

ρ(Xα) := κα This is a refinement of a homomorphism originally considered in [9] Taking the transpose

of ρ, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a K0+(CRFS)– graded version of the Hopf algebra of quasisymmetric functions

Acknowledgements: I would like to thank Dirk Kreimer for many valuable conversa-tions, and the referee for their helpful comments

2 Recollections on CRFS

We briefly recall the definition and necessary properties of the category CRFS, and cal-culate its Grothendieck group For details and proofs, see [6] While [6] treats the case of uncolored trees, the extension of the results to the colored case is immediate Please note that the notion of labeling in [6] and coloring used here are distinct

2.1 The category CRFS

We begin by reviewing some notions related to rooted trees Let S be a set For a tree

T , denote by V (T ), E(T ) the vertex and edge sets of T respectively

Definition 2.1 1 A rooted tree colored by S is a tree T , with a distinguished vertex r(T ) ∈ V (T ) called the root, and an map l : V (T ) → S An isomorphism between two trees T1, T2 labeled by S is a pair of bijections fv : V (T1) ≃ V (T2), fe : E(T1) ≃ E(T2) which preserve roots, colors, and all incidences - we often refer to this data simply by f Denote by RT (S) the set of all rooted trees labeled by S

2 A rooted forest colored S is either empty, or an ordered set F = {T1, , Tn} where

Ti ∈ RT (S) Two forests F1 = {T1, , Tn} and F2 = {T′

1, , T′

m} are isomorphic if

m = n and there is a permutation σ ∈ Sn, together with isomorphisms fi : Ti ≃ T′

σ(i)

3 An admissible cut of a labeled colored tree T is a subset C(T ) ⊂ E(T ) such that

at most one member of C(T ) is encountered along any path joining a leaf to the root Removing the edges in an admissible cut divides T into a colored rooted forest PC(T ) and a colored rooted tree RC(T ), where the latter is the component containing the root The empty and full cuts Cnull, Cf ull, where

(PC null(T ), RC null(T )) = (∅, T ) and (PC f ull(T ), RC f ull(T )) = (T, ∅) respectively, are considered admissible

4 An admissible cut on a colored forest F = {T1, , Tk} is a collection of cuts C = {C1, , Ck}, with Ci an admissible cut on Ti Let

RC(F ) := {RC 1(T1), , RCk(Tk)}

PC(F ) := PC 1(T1) ∪ PC 2(T2) ∪ ∪ PC k(Tk) Example 2.1 Consider the labeled rooted forest consisting of a single tree T colored by

S = {a, b} with root drawn at the top,

T := a

b

b a

a

b a and the cut edges indicated with dashed lines Then

PC(T ) = b

b a

b and RC(T ) = a

a a

We are now ready to define the category CRFS, of rooted forests colored by S.

Definition 2.2 The category CRFS is defined as follows:

Ob(CRFS) := { rooted forests F colored by S } Hom(F1, F2) := {(C1, C2, f )|Ci is an admissible cut of Fi,

f : RC 1(F1) ∼= PC 2(F2)}

Note: For F ∈ CRFS, (Cnull, Cf ull, id) : F → F is the identity morphism in Hom(F, F )

We denote by Iso(CRFS) the set of isomorphism classes of objects in CRFS

Example: if

F1:= a

b

b

b

a b

a

F2:= a

a b

a b

then we have a morphism (C1, C2, f ), where Ci are indicated by dashed lines, and f is uniquely determined by the cuts

For the definition of composition of morphisms and a proof why it is associative, please see [6] The category CRFS has several nice properties:

1 The empty forest ∅ is a null object in CRFS

2 Disjoint union of forests equips CRFS with a symmetric monoidal structure We denote by F1⊕ F2 the disjoint union of the rooted forests F1 and F2 labeled by S, and refer to this as the direct sum

3 Every morphism possesses a kernel and a cokernel

4 For every admissible cut C on a forest F , we have the short exact sequence

∅ → PC(F )(Cnull ,C,id)

The second property above allows us to define the Grothendieck group of CRFS as

K0(CRFS) := Z[Iso(CRFS]/ ∼ i.e the free abelian group generated by isomorphism classes of objects modulo the relation

∼, where ∼ is generated by differences B − A − C for short exact sequences

∅ → A → B → C → ∅

We denote by [A] the class of A ∈ CRFS in K0(CRFS)

Lemma 2.1 K0(CRFS) ≃ Z⊕|S|

Proof Let •s denote the singleton rooted tree colored s We observe that by repeated application of 2.1, any rooted forest F is equivalent in K0(CRFS) to a sum of such, coming from the vertices of F To say this slightly differently, let v(F, s) denote the number of vertices in F of color s ∈ S, and let ZS denote the free abelian group on the set S, with generators es, s ∈ S Let

Ψ : Z[Iso(CRFS)] → ZS

Ψ(F ) =X

s∈S

v(F, s)es

The subgroup generated by the relations ∼ lies in the kernel of Ψ, so we get a well-defined group homomorphism

Ψ : K0(CRFS) → ZS Now, let

Φ : ZS → K0(CRFS) Φ(X

s

ases) =X

s∈S

as[•s]

Ψ and Φ are easily seen to be inverse to each other

We denote by K0+(CRFS) ≃ N|S| the cone of effective classes in K0(CRFS)

3 Ringel-Hall algebras

We recall the definition of the Ringel-Hall algebra of CRFS following [6] For an intro-duction to Ringel-Hall algebras in the context of abelian categories, see [8] We define the Ringel-Hall algebra of CRFS, denoted HCRF S, to be the Q–vector space of finitely supported functions on isomorphism classes of CRFS I.e

HCRFS := {f : Iso(CRFS) → Q | # supp(f ) < ∞}

As a Q–vector space it is spanned by the delta functions δA, f orA ∈ Iso(CRFS) The algebra structure on HCRF S is given by the convolution product:

f ⋆ g(M) := X

A⊂M

f (A)g(M/A)

HCRF S possesses a co-commutative co-product given by

∆(f )(M, N) = f (M ⊕ N), (3.1)

as well as a natural K0+(CRFS)–grading in which δA has degree [A] ∈ K0+(CRFS) The algebra and co-algebra structures are compatible, and HCRF S is in fact a Hopf algebra (see [6]) It follows from 3.1 that

∆(δA) = X

A ′ ⊕A ′′ ≃A

δA ′⊗ δA ′′, (3.2)

where the sum is taken over all distinct ways of writing A as A′⊕ A′′

4 K0+(CRFS)–graded noncommutative symmetric functions and homomorphisms

Let NCCRF S denote the free associative algebra on K0+(CRFS), i.e the free algebra generated by variables Xα, for α ∈ K0+(CRFS) We give it the structure of a Hopf algebra through the coproduct

∆(Xγ) := X

α+β=γ α,β∈K +

0 (CRF S )

Xα⊗ Xβ, (4.1)

and equip it with a K0+(CRFS)–grading by assigning Xαdegree α This is a K0+(CRFS)– graded version of the Hopf algebra of non-commutative symmetric functions (see [3]) For α ∈ K0+(CRFS), let κα be the element of HCRF S given by

κα := X

A∈Iso(C),[A]=α

δA

Example 4.1 Suppose thatS = {a, b} We then have K0(CRFS) ≃ Z2, and may identify the pair (i, j) ∈ K0+(CRFS) as the class representing forests possessing i vertices colored

“a” and j colored “b” We have for instance

κ(1,1) = δ a

b + δ b

a

+ δ a

⊕

b Theorem 1 The map ρ : NCCRF S → HCRFS determined byρ(Xα) = κα is a Hopf algebra homomorphism

Proof Since NCCRF S is free as an algebra, we only need to check that the κα are com-patible with the coproducts 4.1, i.e that

∆(κγ) = X

α+β=γ α,β∈K +

0 (CRF S )

κα⊗ κβ (4.2)

We have

∆(κγ) = X

A∈Iso(CRF S ) [A]=γ

∆(δA)

A∈Iso(CRF S ) [A]=γ

X

A ′ ⊕A ′′ ≃A

δA ′ ⊗ δA ′′

The result now follows by observing that the term δA ′⊗ δA ′′ occurs exactly once in κ[A′ ]⊗

κ[A ′′ ], which is an element of the right-hand side of 4.2, since [A′] + [A′′] = γ

Let NC denote the “usual” Hopf algebra of non-commutative symmetric functions I.e

NC is the free algebra on generators Yn, n ∈ N, with coproduct defined by

∆(Yn) = X

i+j=n

Yi⊗ Yj

(we adopt the convention that Y0 = 1) Suppose that the labeling set S is a subset of N

We then have group homomorphism

V : K0(CRFS) → N

V (Xases) := Xass, which amounts to adding up the labels in a given forest We can now define an algebra homomorphism

JS : NC → NCCRF S

JS(Yn) := X

α∈K +

0 (CRF S )

V (α)=n

Xα

Lemma 4.1 JS is a Hopf algebra homomorphism

Proof We only need to check the compatibility of the coproduct We have

∆(JS(Yn)) = X

α∈K +

0 (CRF S )

V (α)=n

∆(Xα)

α∈K +

0 (CRF S )

V (α)=n

X

γ 1 +γ 2 =α

Xγ 1 ⊗ Xγ 2

γ,γ ′

V (γ)+V (γ ′ )=n

Xγ⊗ Xγ ′

= JS(∆(Yn))

Composing ρ and JS, we obtain a Hopf algebra homomorphism

ρ ◦ JS : NC → HCRF S

ρ ◦ JS(Yn) = X

A∈Iso(CRF S )

V ([A])=n

δA

which was considered in [9]

5 The transpose of ρ

The graded dual of the Hopf algebra NCCRF S is a K0+(C)–graded version of the Hopf alge-bra of quasi-symmetric functions (see [1]), which we proceed to describe Let QSymCRFS denote the Q–vector space spanned by the symbols Z(α1, α2, , αk), for k ∈ N, and

αi ∈ K+

0 (CRFS) We make QSymCRFS into a co-algebra via the coproduct

∆(Z(α1, , αk)) = 1 ⊗ Z(α1, , αk)

+

k−1

X

i=1

Z(α1, , αi) ⊗ Z(αi+1, , αk) + Z(α1, , αk) ⊗ 1

The algebra structure on QSymCRFS is given by the quasi-shuffle product, as follows Given Z(α1, , αk) and Z(β1, , βl), their product is determined by:

1 Inserting zeros into the sequences α1, , αk and β1, , βl to obtain two sequences

ν1, , νp and µ1, , µp of the same length, subject to the condition that for no i

do we have νi = µi = 0

2 For each such pair ν1, νp, and µ1, , µp, writing Z(ν1+ µ1, , νp+ µp)

3 Summing over all possible such pairs of sequences {ν1, , νp}, {µ1, , µp}

Example 5.1 We have

Z(α1)Z(β1, β2) = Z(α1 + β1, β2) + Z(β1, α1+ β2) + Z(β1, β2, α1)

+ Z(β1, α1, β2) + Z(α1, β1, β2)

One checks readily that the two structures are compatible, and that they respect the

K0+(CRFS)–grading determined by

deg(Z(α1, , αk)) = α1+ + αk The pairing

h, i : QSymCRFS× NCCRF S → Q

determined by

hZ(α1, , αn), Xβ 1 Xβ mi := δm,nδα 1 ,β 1 δα nδβ m

makes QSymCRFS and NCCRF S into a dual pair of K0+(CRFS)–graded Hopf algebras I.e

ha ⊗ b, ∆(v)i = hab, vi h∆(a), v ⊗ wi = ha, vwi

This implies that QSymCRFS is isomorphic to the graded dual of NCCRF S Passing to graded duals, and taking the transpose of the homomorphism ρ, we obtain a Hopf algebra homomorphism

ρt : H∗CRFS → QSymCRFS

As shown in [6], H∗CRFS is isomorphic to the Connes-Kreimer Hopf algebra on colored trees (see [5])

We proceed to describe ρt Let {WA, A ∈ Iso(CRFS)} be the basis of H∗CRFS dual to the basis {δA} of HCRFS

Theorem 2

ρt(WA) =X

k

X

V 1 ⊂ ⊂V k =A

Z([V1], [V2/V1], , [Vk/Vk−1]),

where the inner sum is over distinct k–step flags

V1 ⊂ V2 ⊂ ⊂ Vk= A, Vi ∈ Iso(CRFS)

Proof We have

ρt(WA)(Xα 1 Xα k) = N(A; α1, , αk), where N(A; α1, , αk) is the coefficient of δA in the product κα 1κα 2 καk It follows from the definition of the multiplication in the Ringel-Hall algebra that this is exactly the number of flags

V1 ⊂ V2 ⊂ Vk, where [V1] = α1, [V2/V1] = α2, , [Vk/Vk−1] = αk

Example 5.2 Let S = {a, b} as in example 4.1 Using the notation introduced there, we have

ρtW a

b a



= Z((2, 1)) + Z((0, 1), (2, 0)) + Z((1, 0), (1, 1))

+ Z((1, 1), (1, 0)) + Z((1, 0), (0, 1), (1, 0)) + Z((0, 1), (1, 0), (1, 0))

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